Bose-Einstein condensation Bose-Einstein condensation in relativistic quasi-chemical equilibrium systemin relativistic quasi-chemical equilibrium system

　 　 --- --- from color superconductivity to diquark BECfrom color superconductivity to diquark BEC--------

1) QCD phase diagram

2) Introduction to Color superconductivity (CSC)

3) Pair fluctuation above Tc in strong coupling region (=low density)

Contents:

・ Cooper instability in quark matter and BCS theory

・ Patterns of symmetry breaking in CSC

・ Effects of quark-pair fluctuation above Tc ---Pseudo Gap, specific heat,…

・ Diquark formation and its Bose-Einstein Condensation (BEC)

Nakano, Eiji (NTU)

4) Summary and outlook

1) QCD phase diagram1) QCD phase diagram

150~170MeV

Color Superconductivity(CSC)Hadrons

T

Chiral Symm.Broken

0

Tc~100MeV

Hadronic excitations in QGP phase•Soft mode of chiral transition - Hatsuda, Kunihiro.•qq quasi bound state - Shuryak, Zahed; Brown, Lee, Rho•Lattice simulations – Asakawa, Hatsuda; etc.

Pre-critical region of CSC

•large pair fluctuations – Kitazawa, Kunihiro•Crossover from BCS to BEC – Nishida, Abuki

2) Introduction to Color superconductivity (CSC)2) Introduction to Color superconductivity (CSC)

Basic concept of CSC is quite similar to BCS theory :

Electron-phonon system Quark-gluon system

gluonphonon

Attractive interaction comes from background lattice vibration, phonon, Not from the gauge field.

Attractive interaction exists in the elementary level, exchange of gauge bosons.

Attractive interaction causes an instability of fermion many-body system.

Cooper instability (T=0)

T-matrix (two-particle collective mode near F.S.)

= +

VV

VV

The existence of the pole does not imply the bound state of two fermions, but instability of normal phase against the two-particle collective excitation with zero energy ----- a condensation of pair field.

BSC state : superposition of two-particle occupied and absent states, there is no singularity (pole) any more. Break down of U(1) symmetry

(Phase transition to superconductor!)

O.P. :

FKK

K

2-body problem in medium

l

ll V

V

1 has a pole for arbitrary weak attractive interaction at T=0.

Attractive channels in quark matter

[３ ]c×[３ ]c＝ [３ ]c＋ [６ ]cgluon

Quark-quark interaction is mediated by gluons, which has attractive channels for color anti-symmetric quark pairs.

flavor and spin are determined so as to antisymmetrize the two-quark state: Flavor anti-symmetric,

Hadron

2SC phaseCFL phase

T

Order parameter of CSC:

2-flavor case, 2SC phase

3-flavor case, CFL phase

2nd order 1st order

Similar to standard model

Higgs diquark

・ Symmetry breaking pattern and RG analysis shows IR stable fixed point.・ GL analysis shows Type-II superconductivity (fluctuation of gluon is negligible)

Hadron 2SC phase

T

Critical phenomena of 2SC = 2Critical phenomena of 2SC = 2ndnd order phase transition, order phase transition, because because

Determination of Tc : Thouless Criterion Determination of Tc : Thouless Criterion

2nd order

Thus, one can employ the Thouless criterion for 2nd order phase transition: Singularity of T-matrix at finite T gives Tc.

2

2

( 0)

at Tc

ThermodynamicPotential

CT T

()

Dominance of pair fluctuation

T

0

strong coupling!

Mean field approx.works well.

Nature of CSC

There exists large quantum fluctuation of pair field above Tc.

Large quantum fluctuation ( to be Diquark composite)

Large coherence of pair field

weak coupling

Short coherence length N-1/3

3) Pair fluctuation above Tc 3) Pair fluctuation above Tc in Strong coupling region (= low density region)in Strong coupling region (= low density region)

Study on pair fluctuation above Tc (by Kitazawa, Kunihiro)

1) Appearance of Pseudo-Gap

2) Precursory phenomena--- heat capacity, electric conductivity

Hadron 2SC phase

T

Tc

This region

e.g, T-matrix Approximation in NJL modele.g, T-matrix Approximation in NJL modele.g, T-matrix Approximation in NJL modele.g, T-matrix Approximation in NJL model

0

1( , )

( , ) ( , )nn n

Gi i

iG

k

kk

Quark Green function :

( , )n k , n mi i k q

, miq

30

3

q( , )

(2 )( , ) mn m

m

dT G

q qk

A AIH In Random Phase Approximation, (Kitazawa, Kunihiro, PRD2003)

( , ) CG k 1

1

( , )CG Q k

T-matrix (pair collective mode) :

( , )nQ k, n mi i k p

, mip

Gc

k

2 2sgn( ) ( )k k

Quasi-particle energy:

2 2( )

d k

dk k

2( )dk

N kd

( )N

2Density of State:

Quarks in BCS Theory (below Tc)Quarks in BCS Theory (below Tc)

Characterized by finite O.P. :

Gap opens around the Fermi surface!Gap function

=

Pair fluctuation effect (above Tc)Pair fluctuation effect (above Tc)

Quasi-particle energy:

1Re ( , ) 0G p

Dispersion relation:

×1.5

F.S.

Density of State:

0 01( , ) Tr Im ( , )

4RG k k

30

3( ) ( , )

(2 )

dN

kk

Spectral function:

Characterized by zero O.P. :

Density of State:

30

3( ) ( , )

(2 )

dN

kk

Free quark

Pseudo Gap

(Quasi) Level repulsion of spectrum(Quasi) Level repulsion of spectrum(Quasi) Level repulsion of spectrum(Quasi) Level repulsion of spectrum

GC=4.67GeV-2 Fluctuation causes a virtual mixing between quarks and holes

k

nf ()

kF

hole

paritcle

The pseudogap survives up to =0.05~0.1 ( 5~10% above TC ).

Numerical Result : Density of StateNumerical Result : Density of StateNumerical Result : Density of StateNumerical Result : Density of State

( )

( )free

N

N

Enhancement of cV ~-1/2 above Tc.

CV

/107

free fl.C C

Tc

free (BCS approx.)

from collecitve mode

Fluctuation effect on Specific heatFluctuation effect on Specific heat

2 2

2 2. .

/

/

free free

fl fl

C Td dT

C Td dT

free fl.VC C C

Abrupt delay of cooling in compact-star evolutions.

Quark matter core

Summarizing the points so far,

T

Weak Coupling (High density)

Strong Coupling (Low density)

CSC(= CFL)

Pair Fluctuation develops

No Pair Fluctuation

CSC(= 2SC)

Pseudo Gap (pair fluc.)vanishes

Thouless Criterion

Tc

Crossover to BEC

Diquark BEC

Dissociation temp.

Diquark-quark mixture

T_pair fluc.

?

T

baryon

Confinement Phase

Quark Fermi-degeneracy.

Attractive channel.+

(color-3 , flavor-1 , total J=0)

SU(3)SU(3)

*

SU(2)SU(2)cc cc

2-flavor Color 2-flavor Color Superconductivity (2SC)Superconductivity (2SC)

Large quark-pair fluctuation with asymptotic freedom.

Bose statistics of diquark.

Diquark Bose-EinsteinDiquark Bose-EinsteinCondensationCondensation

loosely-boundCooper pair

tightly-bounddiquark cluster

BEC-BCSBEC-BCScrossovercrossover

QGP

Quasi-Chemical Equilibrium Theory.Quasi-Chemical Equilibrium Theory.

Properties of diquark-BECdiquark-BECCritical temperature (Tc) .Density profile.Residual Interaction between diquaks.

Contribution from Pair fluctuation(Diquark propagator)

Free quark part

We obtain the equation for the Baryon number density:

: Bose distribution : Fermi distribution

: scattering phase shift defined by

Thermodynamics with pair fluctuation

Derivative of the phase shift in dilute limit:

For sufficiently large coupling, there appear resonant or bound states below the Fermi Sea in addition to scattering states near the Fermi Sea.

Thus the Baryon number density becomes,

:measured from

which shows a chemical equilibrium between two quark and diquark composite.

(= spectral function of T-matrix)

From the above argument, we reached an ancient approach to superconductivity:

Quasi Chemical Equilibrium Theory (QCET) ( Schafroth, Butler, Blatt, 1956)

which is revived as a strong coupling theory of CSC.

The number conservation:

Chemical equilibrium between quark and diquark:

We have only two parameters, constituent quark mass : and diquark-composite mass :

: Diquark as resonant state

: Diquark as bound state

For a fixed Baryon number N_B,

gives Tc for Diquark BEC. (This is nothing but Thouless criterion.)

(These masses are originally determined from QCD.)

Application for QCD with (u, d) quark matter

Diquark molecules with 2SC2SC-type paring state (color-3 , flavor-1 , total J=0) :*

q + q (qq) = Dchemical eq.chemical eq.

2SC2SC

Other less attractive quark-channels (color-3 , flavor-3 , total J=1)has been recently suggested.

*E.Nakano, et.al.,PRD 68,105001(2003)D.H.Rischke, et.al.,PRD 69,094017(2004)

One-BEC theoremOne-BEC theoremMulti-component fermionic matter ; (color, flavor, spin, etc)

Composite-boson molecules with various channels ;

BEC-singularity occurs onlyonly on the ground state of the most stable channel ( ) :

F1, F2 , F3 ,B1, B2 , B3 ,

F+F B( mB1 < mB2 < mB3 < )

B1 B mB1

Diquark-BEC is ‘homogeneous’ (= no-coexisting state). c.f. Color-Superconductivity** Anti-diquark cannot be condensed into BEC with positive baryon number density ( ).0 d

One-BEC Theorem

Multi-component fermionic matter ; (color, flavor, spin, etc)

Composite-boson molecules with various channels ;F1, F2 , F3 ,

B1, B2 , B3 ,F+F B ( mB1 < mB2 < mB3 < )

Total fermion number conservation.

A composite boson is constructed by 22 fermions( 2-body correlations are included in the theory ).

Helmholtz free-energy density with above constraint ;

Minimum condition of free-energy ( , ) gives ;

for( )

Chemical equilibrium condition.

2

2

is free-energy for one particle.

If , system loses free-energy from fermionic degrees of freedom and gains free-energy from bosonic degrees of freedom.

F+F B

Chemical eq. means the balance between these lose and gain of free-energy.

(one chemical potential control the whole system).

**

positive norm condition.

One-BEC Theorem

Shared chemical potential ( ) must be smaller than any ground state of boson spectra ;

Constraint given by B1 is most severe !

B1 bosons

B bosonsi i 1( )

If T is lowered, will increase to maintain the conserved number density and finally saturatesaturate at .

: BEC-singularityBEC-singularity

At thermodynamical limit ( V ), gives the macroscopic contributions.

[BB11-BEC-BEC]

: no BEC-singularityBEC-singularity

can always be neglected for V . The lightest composite bosons can only be condensed

to the BEC states (one-BEC theorem).

Bose-Einstein condensation occurs only on a ground state of whole boson spectra in the system.

Diquark Bose-Einstein Condensation

Diquark molecules with 2SC2SC-type paring state (color-3 , flavor-1 , total J=0) :*

q + q (qq) = Dchemical eq.chemical eq.

2SC2SC

1) Total baryon number conservations law :

3 3 3 12 2 2 1q qb d dn n n n

2) Chemical equilibrium condition :

2 q d

c f s color-3 , flavor-1 , J=0

Composite-factor

Environmental parameters :

Mass parameters :

, )( b T( , )q dm m

Critical Temperature with Various Mass Values ; ( , )q dm m

Tc* will increase with b large.

No-BEC phase will decrease with b large.*“many-body effect” of BEC.

Mass phase diagramMass phase diagram determines the occurrence of BEC at a certain temperature for various mass values ; .( , )q dm m*

Mass Phase DiagramMass Phase DiagramNo-BEC-

phase

Mass Phase Diagram

T > 0

T = 0 0 00 finitefinite

0

Single bose gas case.

region 1 ( bound state case ; )

BEC-phaseWith the manifest advantage of binding energy,all quarks are combined into diquarks at T 0and condensed into the ground state.

q + q D(0)

loss of kinetic energywith Pauli-blocking.

loss of resonance energy ;

Small - ( loss of resonance energy is small ).

Large - ( loss of resonance energy is large ).

BEC-phase

no BEC-phase

region 2 ( resonant state case ; )

Critical Temperature with Various Mass Values ; ( , )q dm m

* with fixed corresponds to strong coupling limit. (Strong interaction may change the mass of composites with relativistic-energy scale.)

(strong coupling limit) gives .

0『　　　　 』 qm

* 0dm 『　　　　 』 『　　　　 』Tc

0dm If 2 0q d : no-thermal part.

All the conserved baryon number density are bound into diquarks and condensed into the ground state.

1) Non-relativistic RPA gives the saturation of Tc at the strong coupling limit.c.f. P.Nozieres and S.Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985)

2) Photon has no BEC.

Deconfinement phase transition and Chiral phase transitionoccurs at the same point.

< Lattice-QCD simulation for = 0 : T.Celik, et.al., Nucl. Phys. B256. 670 (1985) >

Chiral symmetry restoration cannot precede deconfinement.< Instanton-induced interaction : E.V.Shuryak, Nucl. Phys. B203, 140 (1982) >

< QCD sum rules : A.I.Bochkarev, et.al., Nucl. Phys. B268, 220 (1986) >

These 2 are same phase transition.< glueball-sigma mixing : Y.Hatta, et.al., Phys. Rev. D69, 097502 (2004) >

*

T T

150 MeV 150 MeV

02 020 0

Case 1 : remainig Chiral sym. breaking Case 2 : Chiral sym. restored

Deconfinement phase transition Deconfinement phase transition

Quark Mass & Diquark Mass mq md

*

*

Chiral Chiral symmetry breakingsymmetry breaking

Chiral Chiral symmetry breakingsymmetry breaking

Deconfinement phase transition Deconfinement phase transition

QCD Phase Diagram

* Tc ~ 100MeV : comparative with Tc of Color Superconductivity.c.f. K.Rajagopal and F. Wilczek, hep-ph/0011333 (2000)

* Tc < Tccase1 case2

with light diquark mass .dmcase2

Case 1 : remainig Chiral sym. breaking Case 2 : Chiral sym. restored

diquark-BECdiquark-BEC diquark-BECdiquark-BEC

* Current quark mass in case 2 is too small relative to the energy scale of diquark-BEC.

Density Profile

*Diquarks will condense into the ground state ( D ) below Tc (2nd-order phase transition).

(0)

‘ Saturation of μ’

* ‘Dissociation’ of diquark-molecules is strongly suppressed.

No anti-particle case.

High-T Region of Density Profile

22 22

11 11

T

*Quantum Statistics (Fermi or Bose) gets more important for high-T region with pair creation.

Boltzmann statistics only appears around moderate temperature region.

There is no dissociation for both meaning of baryon number densityand particle (anti-particle) number density, without following effects in QCET, *

1) Asymptotic freedom 2) Medium effect (Pauli-blocking).

1/ 2=

Compositeness :

Symmetry : ,Statistics :

At least, we might have to introduce a energy cut-off of O(B.E.) in diquark density.

Effect of Diquark Interactions

Diquarks are colored objects (color-3 ), not singlet.

Diquarks can scatter into different states through the residual interaction (gluon-exchange).

Strong-coupling limit may not correspond to free bose gas,but (strongly) correlated bose gas system in QCD (?)

**

T is very sensitive for the residual interactions between bosonsin general BEC study.

* cBEC

Effect of diquark interactions is not included in Gaussian-type approximation like RPA.

*c.f. P.Nozieres and S.Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985).

Effect of m lowers Tc, up to freeze-out.

-transition of liquid He, =2.17K (c.f. =3.1K).*

Effect of ‘density homogenization’ rises up Tc by ~10%.

P.Gruter D.Ceperley, and F.Laloe, PRL 79,3549(1997).

H.T.C.Stoof, PRA 45, 8398(1992).

G.Baym, J.P.Blaizot, PRL 83,1703(1999).

Effect of μ does not change Tc at all in single bose gas case.*A.L.Fetter and J.D.Walecka, Quantum Theory ofMany-Particle System (McGraw-Hill, New York, 1971).

4 TT

Phase diagram obtained from QCET

(This effect never appears in T-matrix appr. (=RPA) )

We expect decrease of Tc due to the quark-pair fluctuation and diquark-diquark interaction. (=Diquark composite)

0

90 MeV

BECHadron Phase

Tc~100 MeV for CSCTc~ 30 MeV for BEC

Turn on diquark-diquark int.

enhances Diquark mass.(decreases Tc)

Effect of Diquark Interactions

3-component vector field ; color-3 diquak*

*Contact -term describes the diquark-diquark scattering effect.

*Gross-Pitaevski approach

Higher-order scattering terms ( ) are renormalized into two-body interaction, as usual in nucleon case. J.D.Jackson, Annu. Rev. Part. Sci. 33, 105 (1983)

Effective Lagrangian

Interaction energy :Interaction energy :

HI

2 21( ) ( )

2 dm † † †Leff

MF approximation

dn dn,

Single particle energy spectra of diquark :Single particle energy spectra of diquark :

-Renormalization*

* * -renormalization does not change Tc of BEC at allin single Bose gas case. A.L.Fetter and J.D.Walecka, Quantum Theory of Many-Particle Systerms

All the information about interaction is fully lost in BEC condition.

* * -renormalization gives the leading order of interaction effectin equilibrium system.

chemical equilibriumchemical equilibrium

q+q DIsolated quarks feel the effect of O( ) .

Diquark number density :Diquark number density :

d *

* Quasi-chemical eq. theory (free-q and free-D)*+ -renormalization ; d d*

free

Tc with Diquark Interaction

Residual interaction between color-3 diquarks is estimated from the mass-difference of nucleon and with the assumption of quark-hadron continuity :

*

30 50 0 : repulsive: repulsiveJ.F.Donoghue and K.S.Sateesh, Phys. Rev. D38, 360 (1988)

The positiveness of is also suggested by using P-matrix method.R.L.Jaffe and F.E.Low, Phys. Rev. D19. 2105 (1979)

Residual diquark-diquark interaction will lower the Tc of diquark-BECby ~50% from that in non-interacting case.*Gaussian-type approximation like Nozieres-Schmitt-Rink approachmay not be able to describe the strong-coupling region in QCD ; diquark-BEC (?)* `

~

SummaryDiquark Bose-Einstein condensation is investigated with Quasi-Chemical Equilibrium theory.

Diquark-BEC is ‘homogeneous’ (= no-coexisting state).

Anti-diquark cannot condense into BEC with positive baryon number density.

(strong coupling limit) gives ; relativistic effect .0dm 『　　　　 』 『　　　　 』Tc

Tc ~ 100MeV ; comparative with Tc of Color Superconductivity.BEC

‘Dissociation’ is strongly suppressed with pair creation.

Quantum statistics still remains for T with pair creation.

Residual diquark-diquark interaction lowers Tc by ~50%.( less applicability of Gaussian-type approximation ?)

Future Work

The effect of 3-body correlations (q-D, q-q-q) for the phenomena of 2-body clustering matter.

Summary

We viewed the quark-pair correlation (fluctuation) at finite density from weak (high density) to strong (low density) regimes.

Outlook

weak

strong

1) Color superconductivity

2) Pair fluctuation develops above Tc ・ Pseudo gap phenomena ・ Enhance of specific heat

3) Formation of Quasistable diquarks (= quantum fluctuation) ・ Crossover to Diquark BEC

Observable consequences in experiments or in astrophysical observations, e.g., effects on dilepton or neutrino production rate, and response to external magnetic field.

I thank Mr. Nawa (Dept. of Phys. in Kyoto Univ.) for his close collaborations.

High Tc !High Tc !

Feshback resonance scattering

Observation of di-fermion BECObservation of di-fermion BEC

Interaction strength can be controlled artificially!

Fermion Atoms in trapping potential.

Softening of Pair FluctuationsSoftening of Pair FluctuationsSoftening of Pair FluctuationsSoftening of Pair Fluctuations

Dynamical Structure Factor

=0.05

The peak grows from ~ 0.2 electric SC ： ~ 0.005

1 1( ) I (m )1

Se

kk

= 400 MeV

c

c

T T

T

= 400 MeV

Pole of Collective Mode

1 1 ( , ) 0CG Q kpole:

c

c

T T

T

The pole approaches the origin as T is lowered toward Tc.

(the soft-mode of the CSC)

stronger diquark coupling GC

Diquark Coupling DependenceDiquark Coupling DependenceDiquark Coupling DependenceDiquark Coupling Dependence

GC ×1.3 ×1.5

= 400 MeV=0.01

Numerical results in QCETa

The explicit form of the equation

Dispersion of quark and diquark are given by :

where

Upper bound of : RHS

0